On we consider the Schrödinger operator
where and is a positive function (“potential”).
Let be the heat semigroup associated with to . In the talk we shall consider the Hardy space
which is a natural substitute of in harmonic analysis associated with . Our main interest will be in showing that elements have decompositions of the type , where and (“atoms”) have some nice properties.
In the classical case on an atom is a function for which there exist a ball such that
We shall see that for we can still prove some atomic decompositions, but the properties of atoms depend on the dimension d and the potential .